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Question-202016

Question Number 202016 by sonukgindia last updated on 18/Dec/23 Answered by MM42 last updated on 18/Dec/23 $${x}^{\mathrm{55}} −\mathrm{1}=\mathrm{253}×\mathrm{8}{k}\Rightarrow“{x}''\:{is}\:{odd} \\ $$$$\left({x}−\mathrm{1}\right)\left({x}^{\mathrm{54}} +{x}^{\mathrm{53}} +…+\mathrm{1}\right)=\mathrm{253}×\mathrm{8}{k} \\ $$$${x}−\mathrm{1}=\mathrm{8}{k}'\Rightarrow{x}\overset{\mathrm{8}} {\equiv}\mathrm{1}…

Question-202017

Question Number 202017 by sonukgindia last updated on 18/Dec/23 Answered by Mathspace last updated on 18/Dec/23 $${f}\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \frac{{sin}\left({nx}\right)}{\mathrm{2}^{{n}} }\:\Rightarrow \\ $$$$\int_{\mathrm{0}} ^{\pi} {f}\left({x}\right){dx}=\int_{\mathrm{0}} ^{\pi}…

Question-201949

Question Number 201949 by sonukgindia last updated on 16/Dec/23 Answered by Frix last updated on 17/Dec/23 $${n}^{{k}} \equiv{n}\mathrm{mod2024};\:{n}\in\left\{\mathrm{529},\:\mathrm{737},\:\mathrm{760},\:\mathrm{1265},\:\mathrm{1288},\:\mathrm{1496}\right\} \\ $$$$\Rightarrow \\ $$$$\Sigma{n}^{{n}} \equiv\mathrm{3mod2024} \\ $$…

Question-201956

Question Number 201956 by sonukgindia last updated on 16/Dec/23 Commented by mahdipoor last updated on 16/Dec/23 $$\mathrm{5}^{\mathrm{2}^{\mathrm{117}} } −\mathrm{1}=\left(\mathrm{5}^{\mathrm{2}^{\mathrm{116}} } −\mathrm{1}\right)\left(\mathrm{5}^{\mathrm{2}^{\mathrm{116}} } +\mathrm{1}\right)= \\ $$$$\left(\mathrm{5}^{\mathrm{2}^{\mathrm{115}}…

Question-201906

Question Number 201906 by sonukgindia last updated on 15/Dec/23 Answered by AST last updated on 15/Dec/23 $${sin}\left({x}\right)=\frac{{s}}{{h}}…\left({i}\right) \\ $$$$\left({ii}\right)…\frac{{sin}\left(\mathrm{2}{x}\right)}{{h}}=\frac{{sin}\left({x}\right)}{\mathrm{2}{s}}\Rightarrow\mathrm{2}{sin}\left({x}\right)=\frac{{sin}\left({x}\right)}{{sin}\left(\mathrm{2}{x}\right)} \\ $$$$\Rightarrow{sin}\left(\mathrm{2}{x}\right)=\frac{\mathrm{1}}{\mathrm{2}}\Rightarrow{x}=\mathrm{15}° \\ $$ Answered by…

Question-201898

Question Number 201898 by sonukgindia last updated on 15/Dec/23 Answered by Frix last updated on 15/Dec/23 $${I}=\mathrm{4}\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\frac{{dx}}{\mathrm{1}+\mathrm{3sin}^{\mathrm{2}} \:{x}}\:\overset{{t}=\mathrm{tan}\:{x}} {=}\:\mathrm{4}\underset{\mathrm{0}} {\overset{\infty} {\int}}\frac{{dt}}{\mathrm{4}{t}^{\mathrm{2}} +\mathrm{1}}= \\…

Question-201888

Question Number 201888 by MrGHK last updated on 15/Dec/23 Answered by namphamduc last updated on 15/Dec/23 $${S}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\psi\left({n}+\mathrm{2}\right)}{\left({n}+\mathrm{2}\right)^{\mathrm{2}} }=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\psi\left({n}+\mathrm{1}\right)}{\left({n}+\mathrm{1}\right)^{\mathrm{2}} }=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{H}_{{n}}…